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Coin Toss Probability Formula and Examples

Coin toss probability is an excellent introduction to the basic principles of probability theory because a coin has a mostly equal chance of landing heads or tail. So, a coin toss is a popular and fair method of making an unbiased decision. Here is a look at how coin toss probability works, with the formula and examples.

• When you toss a coin, the probability of getting heads or tails is the same.
• In each case, the probability is ½ or 0.5. In other words, “heads” is one of two possible outcomes. The same is true for tails.
• Find probability of multiple independent events by multiplying the probability of individual events. For example, the probability of getting heads and then tails (HT) is ½ x ½ = ¼.

The Basics of Coin Toss Probability

A coin has two sides, so there are two possible outcomes of a fair coin toss: heads (H) or tails (T).

Coin Toss Probability Formula

The formula for coin toss probability is the number of desired outcomes divided by the total number of possible outcomes. For a coin, this is easy because there are only two outcomes. Getting heads is one outcome. Getting tails is the other outcome.

P = (number of desired outcomes) / (number of possible outcomes) P = 1/2 for either heads or tails

The probability of getting either heads or tails (2 possible outcomes) is 1. In other words, when you toss a coin you are pretty much guaranteed to get either heads or tails.

P = 2/2 = 1

Getting heads or tails on a coin are mutually exclusive events . If you get heads, you don’t get tails (and vice versa). Another way of calculating the probability of two mutually exclusive events is adding their individual probabilities. For one coin toss:

P(heads or tails) = ½ + ½ = 1

Probability for Multiple Coin Tosses

If you toss a coin more than once and want the probability of a specific outcome, you multiply the probability values of each toss. This works when the tosses are independent events . What this means is the outcome of the second toss (or third, etc.) is not dependent on the outcome of the first toss (or any other previous or subsequent toss).

For example, let’s calculate the probability of getting heads, heads, tails (HHT):

P(HHT) = ½ x ½ x ½ = ⅛

Coin Toss Probability Example Problems

Coin toss problems usually are word problems. The key is understanding what the problem is asking.

For example, calculate the probability of tossing a coin twice times and getting at least one “heads”.

First, write down all the possible outcomes of randomly tossing a coin three times:

HH, HT, TH, TT

There are four possible outcomes.

Next, determine how many of these outcomes are “favorable outcomes” or ones that meet the criteria in the problem. There are three outcomes where at least one toss has a “heads” result.

Now, perform the calculation:

P = favorable outcomes / total outcomes P (at least one H) = 3/4 or 0.75

Now, what is the probability of both tosses showing the same face? In other words, what is the chance of both tosses showing heads or both showing tails?

Again, you have four possible outcomes. There are two favorable outcomes (HH or TT).

P (both heads or both tails) = 2/4 = 1/2 or 0.5

What Is a Fair Coin?

A “fair coin” is one which has an equal probability of landing heads or tails in a coin toss. In contrast, an unfair coin is one which is weighted or filed so that it has a greater chance of landing on one side than the other.

In practice, most coins are not totally fair because the raised metal slightly favors one side (on the order of 0.49 to 0.51). Also, for an ordinary person, there is a slight bias that favors catching a coin in the same orientation as how it was thrown (0.51). Skilled conjurers and gamblers can toss or catch a coin so that it lands with considerable bias, even if the coin is fair.

There is also a slight chance of a coin landing on its edge. For example, an American nickel lands on its edge about 1 in 6000 tosses.

Randomness and Probability

Even though a fair coin has even odds of a heads or tails result, the outcome is random. So, if you toss a coin twice, probability calculates you only have a 1 in 4 chance of getting HH. If you repeat the process and toss the coin two more times, you can get different results. The probable outcome becomes more likely the more times you repeat the process.

With this in mind, do you think a coin is biased if it is tossed a certain number of times and 3/4 (75%) of the time it was heads? The answer is that you cannot make a determination of fairness, because you don’t know whether the coin was tossed four times or four thousand times! If, however, you know the number of tosses, you have a real sense of whether or not a coin is fair.

• Ford, Joseph (1983). “How random is a coin toss?”. Physics Today . 36 (4): 40–47. doi: 10.1063/1.2915570
• Kallenberg, O. (2002) Foundations of Modern Probability (2nd ed.). Springer Series in Statistics. ISBN 0-387-95313-2.
• Murray, Daniel B.; Teare, Scott W. (1993). “Probability of a tossed coin landing on edge”. Physical Review E . 48 (4): 2547–2552. doi: 10.1103/PhysRevE.48.2547
• Vulovic, Vladimir Z.; Prange, Richard E. (1986). “Randomness of a true coin toss”. Physical Review A . 33 (1): 576–582. doi: 10.1103/PhysRevA.33.576

Related Posts

Tossing a Coin

There are always two sides to a coin- Heads and Tails.

Have you ever made an important decision by flipping a coin?

In American football, the captain of each team chooses a side of a coin and then the referee tosses the coin.

The captain who predicts the toss correctly decides about the goal which his team would defend.

Let us see how this works and what is the probability of each side of a coin.

Lesson Plan 

What Are the Possible Events that Can Occur When A Coin Is Tossed?

Every coin has two sides: Head and Tail

We denote Head as H and Tail as Tail.

When a coin is tossed, either head or tail shows up. 

The set of all possible outcomes of a random experiment is known as its  sample space . Thus, if your random experiment is tossing a coin, then the sample space is {Head, Tail}, or more succinctly, { H ,  T }.

If the coin is  fair , which means that no outcome is particularly preferred, or every outcome is  equally likely , then we know that for a large number of tosses, the number of Heads and the number of Tails should be roughly equal. That is, the number of Heads should be roughly 1/2 of the total number of tosses, and so should be the number of Tails. This numerical quantity of 1/2 can be used as a  measure of likelihood , or  probability .

What Do You Mean by Tossing A Coin Probability?

Tossing A Coin Probability is the chance of each side of the coin to show up.

The action of tossing a coin has two possible outcomes: Head or Tail. You don’t know which outcome you will obtain on a particular toss, but you do know that it will be either Head or Tail (we rule out the possibility of the coin landing on its edge!).

Contrast this with a science experiment. For example, if your experiment is to drop an object, you know the outcome for sure: the object will fall towards the ground. However, tossing a coin is a  random experiment , as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

The general formula to determine the probability is:

\(\text{Probability }= \dfrac{\text{Number of favorable Outcomes}}{\text{Total number of outcomes}}\)

When a coin is tossed, there are only two possible outcomes.

Therefore, using the probability formula

On tossing a coin, the probability of getting a head is:

P(Head) = P(H) = 1/2

Similarly, on tossing a coin, the probability of getting a tail is:

P(Tail) = P(T) = 1/2

Try tossing a coin below by clicking on the 'Flip coin' button and check your outcomes.

Click on the 'Reset' button to start again.

How Do You Predict Heads or Tails?

• If a coin is fair (unbiased), that is, no outcome is particularly preferred, then we cannot predict heads or tails. Both the outcomes are equally likely to show up.
• If the probability of a head showing up is greater than 1/2, then we can predict the next outcome to be a head.
• If the probability of a tail showing up is greater than 1/2, then we can predict the next outcome to be a tail. 

Suppose that you toss a coin 100 times. Which of the following results is  more likely  to occur?

Result-1:  You obtain 95 Heads and 5 Tails

Result-2:  You obtain 48 Heads and 52 Tails

If the coin is a  normal  everyday coin, in which neither side is particularly prone to showing up more than the other side, you would expect that in a large number of tosses, Heads and Tails should show up roughly an equal number of times. This means that out of the two results above, Result-2 seems to be the more likely one, as the number of Heads is roughly equal to the number of Tails, which concurs with the fact that neither Head nor Tail is a preferred outcome.

On the other hand, in Result-1, the number of Heads is much larger than the number of Tails. Clearly, such a result is extremely biased towards Heads, which is not very likely given that Heads and Tails are equally preferred outcomes. Note that we are not saying that Result-1 is impossible. We are only saying that it is  improbable , or  unlikely . In other words, the  likelihood  of Result-1 is much lower than the  likelihood  of Result-2.

The study of Probability enables us to quantify likelihoods. It enables us to answer questions like:  How likely  is Result-2?  How unlikely  is Result-1? And so on.

However, tossing a coin is a  random experiment , as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

Solved Examples

A coin is tossed a certain number of times. The relative occurrence of Heads is 0.75. Can we say that the coin is biased towards Heads?

No, we cannot, because the experiment (tossing the coin) may have been repeated a very small number of times, and thus the relative occurrence in such a scenario will not give the true probability.

Coin-A is tossed 200 times, and the relative occurrence of Tails is 0.47. Coin-B is tossed an unknown number of times, but it is known that the relative occurrence of Heads is 0.50. Which coin is fairer?

It is not possible to comment on the fairness of Coin-B, because the number of times it was tossed is not known. On the other hand, Coin-A seems to be fair, as the relative occurrence of tails over a large number of tosses is almost 1/2.

On tossing a coin twice, what is the probability of getting only one tail?

On tossing a coin twice, the possible outcomes are {HH, TT, HT, TH}

Therefore, the total number of outcomes is 4

Getting only one tail includes {HT, TH}

Therefore, the number of favorable outcomes is 2

Hence, the probability of getting exactly one tail is 2/4 = 1/2

• Relative occurrence   of an outcome   is used to signify the ratio of the number of times that a particular outcome is obtained to the total number of times the random experiment is performed.
• On tossing a coin, the probability of each outcome is 1/2
• P(Head) + P(Tail) = 1

Interactive Questions   

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

• A fair coin is tossed repeatedly. If the tail appears on the first four tosses, then what is the probability of the head appearing on the fifth toss?
• If a coin is tossed thrice successively, what is the probability of obtaining at least one head and at least one tail?

Let's Summarize 

We hope you enjoyed learning about Tossing A Coin with the simulations and practice questions. Now you will be able to easily solve problems on Tossing A Coin math with multiple math examples you learned today.

About Cuemath  

At  Cuemath , our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

Frequently Asked Questions (FAQs)

1. can a coin land on its side   .

A coin can land on its side if it falls against an object such as a box, shoe, etc.

It is unlikely for a coin to land on its side on a flat surface, but we cannot say that it is impossible.

2. Is flipping a coin a good way to make a decision?

Making decisions by flipping a coin helps a person to decide when stuck between two options because each outcome has an equal probability.

3. How do you predict a coin flip?

Tossing a coin is a random experiment, as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

Therefore, we cannot predict a coin flip if the coin is fair.

4. Are coin flips truly random?

Tossing a coin is considered a random event.

According to Newton, if a person flips in a particular manner at a definite speed, then the outcome can be determined.

5. What is the probability of flipping five heads in a row?

On tossing a coin five times, the number of possible outcomes is 2 5

Therefore, the probability of getting five heads in a row is 1/2 5

6. Is flipping a coin a simple random sample?

Simple Random Sample takes a small portion of a large dataset to represent the data.

Tossing a coin is a random experiment and each outcome has equal probability.

Therefore, tossing a coin is a simple random sample.

7. Is a coin toss really 50/50?

On tossing a coin, each outcome has an equal probability and there are two outcomes. 

Therefore, tossing a coin a 50/50. 

8. In a coin toss, is it fairer to catch a coin or let it fall?

On tossing a coin, it is fairer to let the coin fall than catching it because the force of the hands can flip it.

9. What are the odds of flipping three heads in a row?

On tossing a coin three times, the number of possible outcomes is 2 3

Therefore, the probability of getting five heads in a row is 1/2 3

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COIN TOSSING EXPERIMENT

Coin tossing experiment always plays a key role in probability concept. Whenever we go through the stuff probability in statistics, we will definitely have examples with coin tossing.

Sample Space

When a coin is tossed, there are two possible outcomes.

They are 'Head' and 'Tail'.

So, the sample space S = {H, T}, n(s) = 2.

When two coins are tossed,

total number of all possible outcomes = 2 x 2 = 4

So, the sample space S  =  {HH, TT, HT, TH}, n(s) = 4.

When three coins are tossed,

total no. of all possible outcomes = 2 x 2 x 2 = 8.

So, the sample space is

S = {HHH, TTT, HHT, HTH, THH, TTH, THT, HTT},

In this way, we can get sample space when a coin or coins are tossed.

In coin toss experiment, we can get sample space through tree diagram also. 

Probability Formula

We can use the formula from classic definition to find probability in coin tossing experiments.

Let A be the event in a random experiment.

n(A) = Number of possible outcomes for the event A

n(S) = Number of all possible outcomes of the experiment

Here "S" stands for sample space which is the set contains all possible outcomes of the random experiment.

Then the above formula will become.

To have better understanding of the above formula, let us consider the following coin tossing experiment. 

A coin is tossed once. 

S = {H, T} and n(S) = 2

Let A be the event of getting head. 

A = {H} and n(A) = 1

P(A) = n(A)/n(S)

Solved Problems

Problem 1 :

A coin is twice. What is the probability of head ?

When two coins are tossed, 

total no. of all possible outcomes = 2 x 2  =  4.

And we have, we have the following sample space.

S  =  {HH, TT, HT, TH} and n(S)  =  4

Letting A be the event of getting head, we have

A  =  {HT, TH} and n(A)  =  2

By the classical definition of probability,

P(B)  =  2/4

P(B)  =  0.50 or 50%.

Problem 2 :

A coin is tossed three times. What is the probability of getting :

(i)  2 heads

(ii)  at least 2 heads

When a coin is tossed three times, first we need enumerate all the elementary events.

This can be done using 'Tree diagram' as shown below :

Hence the elementary events are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

That is, 

S  =  { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }

Thus the number of elementary events n(s)  =  8

(i) 2 heads :

Out of these 8 outcomes, 2 heads occur in three cases namely HHT, HTH and THH.

If we denote the occurrence of 2 heads by the event A and if assume that the coin as well as performer of the experiment is unbiased then this assumption ensures that all the eight elementary events are equally likely.

Then by the classical definition of probability, we have

P(A)  =  n(A) / n(s)

P(A)  =  3/8  

P(A)  0.375 or 37.5%

(ii) at least 2 heads :

Let B denote occurrence of at least 2 heads i.e. 2 heads or 3 heads.

Since 2 heads occur in 3 cases and 3 heads occur in only 1 case, B occurs in 3 + 1 or 4 cases.

P(B)  =  4/8

P(B)  =  0.50 or 50%

Problem 3 :

Four coins  are tossed once. What is the probability of getting at least 2 tails ?

When four coins are tossed once, 

total no. of all possible outcomes  =  2 x 2 x 2 x 2  =  16

S  =  {HHHH, TTTT, HHHT, HHTH, HTHH, THHH, TTTH, TTHT, THTT, HTTT, HHTT, TTHH, HTHT, THTH, HTTH, THHT}

and n(S)  =  16

Let A be the event of getting at least two tails. 

Then A has to include all the events in which there are two tails and more than two tails. 

A  =  { TTTT, TTTH, TTHT, THTT, HTTT, HHTT, TTHH, HTHT, THTH, HTTH, THHT }

and n(A)  =  11

P(B)  =  11/36

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Coin Toss Probability Formula

You might have noticed that before the commencement of a cricket match, a decision is to be made, which team would bat or bowl first. How is this done? You see that the captains of the two teams participate in a coin toss wherein they pick one side of a coin each, that is head or tail. The umpire tosses the coin in the air. The team which wins the toss gets to make the decision of batting or bowling first. This is one of the most common applications of the coin toss experiment.

Why do you think this method is used? This is because the possibility of obtaining a Head in a coin toss is as likely as obtaining a tail, that is, 50%. So when you toss one coin, there are only two possibilities – a head (H) or a tail (L). However, what if you want to toss 2 coins simultaneously? Or say 3, 4 or 5 coins? The outcomes of these coin tosses will differ. Let us learn more about the coin toss probability formula .

Coin Toss Probability

Probability is the measurement of chances – the likelihood that an event will occur. If the probability of an event is high, it is more likely that the event will happen. It is measured between 0 and 1, inclusive. So if an event is unlikely to occur, its probability is 0. And 1 indicates the certainty for the occurrence.

Now if I ask you what is the probability of getting a Head when you toss a coin? Assuming the coin to be fair, you straight away answer 50% or ½. This is because you know that the outcome will either be head or tail, and both are equally likely. So we can conclude here:

Number of possible outcomes = 2

Number of outcomes to get head = 1

Probability of getting a head = ½

\(\begin{array}{l}Probability\;of\;getting\;a\;head = \frac{No\;of\;outcomes\;to\; get\;head}{No\;of\;possible\;outcomes}\end{array} \)

We can generalise the coin toss probability formula:

\(\begin{array}{l}Probability\;of\;certain\;event=\frac{Number\;of\;favourable\;outcomes}{Total\;number\;of\;possible\;outcomes}\end{array} \)

Solved Examples

Question : Two fair coins are tossed simultaneously. What is the probability of getting only one head?

When 2 coins are tossed, the possible outcomes can be {HH, TT, HT, TH}.

Thus, the total number of possible outcomes = 4 Getting only one head includes {HT, TH} outcomes.

So number of desired outcomes = 2

Therefore, probability of getting only one head

\(\begin{array}{l}=\frac{Number\;of\;favourable\;outcomes}{Total\;number\;of\;possible\;outcomes}\end{array} \) \(\begin{array}{l}=\frac{2}{4}=\frac{1}{2}\end{array} \) Question : Three fair coins are tossed simultaneously. What is the probability of getting at least 2 tails?

When 3 coins are tossed, the possible outcomes can be {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Thus, total number of possible outcomes = 8 Getting at least 2 tails includes {HTT, THT, TTH, TTT} outcomes.

So number of desired outcomes = 4

Therefore, probability of getting at least 2 tails =

\(\begin{array}{l}\frac{No\;of\;favourable\;outcomes}{Total\;number\;of\;possible\;outcomes}\end{array} \)

\(\begin{array}{l}=\frac{4}{8}=\frac{1}{2}\end{array} \) To solve more problems on the topic, download BYJU’S -The Learning App.

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